The Finite Element Method

Transcription

1 The Fnte Element Method GENERAL INTRODUCTION Read: Chapters 1 and 2 CONTENTS Engneerng and analyss Smulaton of a physcal process Examples mathematcal model development Approxmate solutons and methods of approxmaton The basc features of the fnte element method Examples Fnte element dscretzaton Termnology Steps nvolved n the fnte element model development

2 INTRODUCTORY REMARKS What we do as engneers? develop mathematcal models, conduct physcal experments, carry out numercal smulatons to help desgner, and desgn and buld systems to acheve a (1) functonalty n (2) most economcal way. Knowng the fundamentals assocated wth each engneerng problem you set out to tackle, not only makes you a better engneer but also empowers you as an engneer. Introducton: 2

3 INTRODUCTORY REMARKS Engneerng s the dscplne, art, and professon of acqurng and applyng techncal, scentfc, and mathematcal knowledge to desgn and mplement materals, structures, machnes, devces, systems, and processes that safely realze a desred objectve. Engneerng s a problem-solvng dscplne, and soluton requres an understandng of the phenomena that occurs n the system or process. Introducton: 3

4 Analyss Analyss s an ad to desgn and manufacturng, and not an end n tself. Analyss nvolves the followng steps: dentfyng the problem and nature of the response to be determned, selectng the mathematcal model (.e., governng equatons), solvng the problem wth a soluton method (e.g., FEM), and evaluatng the results n lght of the desgn parameters. Introducton: 4

5 MODELING OF A Physcal Process Assumptons concernng the system BVP Boundary value problems (equlbrum problems) IVP Intal value problems (tme-dependent problems) Numercal method (FEM,FDM,BEM,etc.) Physcal System Mathematcal Model (BVP, IVP) Numercal Smulatons Laws of physcs (conservaton prncples) FEM Fnte Element Method FDM Fnte Dfference Method BEM Boundary Element Method Computatonal devce Introducton: 5

6 EXAMPLES OF MATHEMATICAL MODEL DEVELOPMENT Objectve: Determne heat flow n a heat exchanger fn Lateral surface and rght end are exposed to ambent temperature, T (a) Rectangular fns Body from whch heat has to be extracted a h L (b) 3D to 2D y z x Convecton, ( qa ) ( ) x qa x + Δ x ( -T ), P = permeter β P T a æax + A ö x+ Δx ( qa) x -( qa) x+ Δx -βpδx ( T - T ) + ρr hç Δx = 0 çè 2 ø l 2D to 1D d dt - ( Aq) -βp( T - T ) + ρrh A = 0, q =-k dx dx Basc Concepts: 6

7 EXAMPLE OF ENGINEERING MODEL DEVELOPMENT (contnued) f, force per unt length d æ dtö - ç Ak + βp( T - T ) = ρrh A dx çè dx ø u= T - T, a= Ak, c= βp, ρrh A = f d æ duö - ç a + cu= f dx çè dx ø Determne: Axal deformaton of a bar x Δx L ( Aσ ) x d ( Aσ) x+ Δx -( Aσ) x + faδx = 0 ( Aσ) + fa= 0, dx du d æ duö σ = Eε, ε =, - ç AE + f = 0 dx dx çè dx ø Δx f ( Aσ ) x + Δ x Basc Concepts: 7

8 Model Problem a = E, A APPROXIMATE SOLUTIONS d du a( x) + c( x) u f ( x) = 0 n Ω = ( 0, L) dx dx du a + b( u u0 ) = P at a boundary pont dx EA f( x) u0 P u(l) unnsulated bar a = ka c= P f( x) β L x, u c b=k Elastc deformaton of a bar x L u= T - T, u = T 0 Heat transfer n a fn Basc Concepts: 8

9 ENGINEERING EXAMPLES OF THE MODEL PROBLEM IN 1-D Flow of vscous flud through a channel d dv x μ f 0 n ( 0, b) dy = Ω = dy uy ( ) = vx ( y), horzontal velocty f = pressure gradent, dp/ dx(constant) μ( y) = flud vscosty Q = shear stress U = velocty of the top surface y b 2 U ³ 0 b x Poseulle flow Couette flow Basc Concepts: 9

10 Exact and Approxmate Solutons An exact soluton satsfes (a) the dfferental equaton at every pont of the doman and (b) boundary condtons on the boundary. An approxmate soluton satsfes the dfferental equaton as well as the specfed boundary condtons n some acceptable sense (to be made clearer shortly). We seek the approxmate soluton as a lnear combnaton of unknown parameters c and known functons f ux ( )» u ( x) = å cf ( x) + f ( x) Approxmate soluton and f 0 N N = 1 0 Actual soluton Approxmaton of the actual soluton over the entre doman Basc Concepts: 10

11 Determnng Approxmate Solutons (contnued) f 1. Suppose that s selected to satsfy the boundary condtons exactly. Then substtuton of u N (x) nto the dfferental equaton d æ du ax N ö - ç ( ) + cxu ( ) N - fx ( ) = 0 dx çè dx ø wll result n a non-zero functon on the left sde of the equalty: d æ du a( x ) N ö - ç + c( x )u N - f( x ) º R( x ) ¹ dx çè dx ø Then c are determned such that the resdual, R(x), s zero n some sense. 0 Basc Concepts: 11

12 METHODS OF APPROXIMATION 1. One sense n whch the resdual, R(x), can be made zero s to requre t to be zero at selected number of ponts. The number of ponts should be equal to the number of unknowns n the approxmate soluton N ux ( ) u = c f ( x) + f0( x) N j j j = 1 fj ( x) and f0( x) are functons to be selected to satsfy the specfed boundary condtons and cj are parameters to be determned such that the resdual s made zero n some sense. Ths way of determnng c s known as the Collocaton method. We obtan N algebrac equatons n N unknown C s R( x )= 0, = 1, 2,, N Basc Concepts: 12

13 Methods of Approxmaton (Contnued) 2. Another approach n whch the resdual, R(x), can be made zero s n a least-squares sense;.e., mnmze the ntegral of the square of the resdual wth respect to C s. Mnmze or J ( c J c 1, c = 2 2,, c 0 ) R R c Ths method s known as the least-squares method. We obtan N algebrac equatons n N unknown C s L N = dx L 0 = R 0 2 dx L 0 R R c dx = 0 Basc Concepts: 13

14 Methods of Approxmaton (Contnued) 3. Yet, another approach n whch the resdual, R(x), can be made zero s n a weghted-resdual sense L 0 = y Rdx, = 1,2,, N ò 0 where y are lnearly ndependent set of functons Ths method s known as the Weghted-Resdual method. We obtan N algebrac equatons n N unknown C s. In general, weght functons ψ are not the same as the approxmaton functons. Varous specal cases are ϕ Petrov-Galerkn Method: Galerkn Method: y ¹f Basc Concepts: 14 y =f

15 WEIGHTED-INTEGRAL FORMULATIONS n the Numercal Soluton of Dfferental Eqs. The approxmaton methods dscussed earler can be vewed as specal cases of the weghted-resdual methods of approxmaton. In partcular, we have Collocaton method Least-squares method Petrov-Galerkn method Galerkn Method y() x = d( x-x ) R y ( x) = c y ( x) ¹ f ( x) y ( x) = f ( x) Basc Concepts: 15

16 Methods of Approxmaton (Contnued) 4. Another approach n whch the governng equaton s cast n a weak-form and the weght functon s taken the same as the approxmaton functon s known as the Rtz method: B( f, u ) = ( f ), = 1,2,, N N The Rtz method s the most commonly used method for all commercal software. In ths method, f 0 satsfes only the specfed essental (geometrc) boundary condtons whle f satsfes the homogeneous form of the specfed essental boundary condtons. The specfed natural (force) boundary condton are ncluded n the weak form. Introducton: 16

17 WORKING EXAMPLE For a weghted-resdual method: f 0 f satsfy the actual specfed BCs satsfy the homogeneous form of the actual specfed BCs For the Rtz method: f 0 satsfy the actual specfed essental BCs f satsfy the homogeneous form of the actual specfed essental BCs d æ duö - ç a - f0 = 0, 0< x< L dx çè dx ø du u(0) = u0, a = P dx x = L Introducton: 17

18 f WORKING EXAMPLE (CONTINUED) For a weghted-resdual method: df P ( 0) = u, a = P f = u + x a dx x= L df f ( 0) = 0, a = 0 f1 = x( 2L-x) dx x= L For the Rtz (or weak-form Galerkn) method: f ( 0) = u, f( 0) = 0 0 0, ( ) f0 = u0 f x = x Introducton: 18

19 BASIC FEATURES OF THE FINITE ELEMENT METHOD (FEM) Dvde whole nto parts (fnte element mesh) Set up the `problem over a typcal part (derve a set of relatonshps between prmary and secondary varables) Assemble the parts to obtan the soluton to the whole Basc Concepts: 19

20 Example 1: Determne the center of mass of a 3D machne part 1. See the smplcty n the complcated (see the geometrc shapes that are smple to dentfy the center of mass). 2. Determne the center of mass of each part. 3. Put the parts together to obtan the requred soluton. N N N mx my mz X = Y = Z = m m m = 1 = 1 = 1,, N N N = 1 = 1 = 1 Basc Concepts: 20

21 Example 2: Determne the ntegral of a functon b I = F( x) a dx F(x) Actual functon a b x Basc Concepts: 21

22 EXAMPLE 2 (contnued) F( x) a1 a2 a + b x, b 2 + b 3 x, x, a x x x x x 2 x x b x+ 1 x+ 1 I = F ( x) dx = ( a + b x) dx x x I I I 2 I3 F(x) Actual functon Approxmaton I1 I I 2 3 a = x b = x4 x 1 x2 3 x Basc Concepts: 22

23 EXAMPLE 2 (contnued) Refned mesh F(x) Actual functon Approxmaton a = x 1 7 b = x x Basc Concepts: 23

24 Approxmaton of a curved surface wth a plane (temperature profle) (represents the soluton) Doman (Trangular element) Basc Concepts: 24

25 Fnte Element Dscretzaton Doman, Ω Elements Boundary, Γ e Ω Nodes (a) Gven doman (b) Fnte element mesh Boundary flux Ω e Doman, Ω h (c) Typcal element wth boundary fluxes (d) Dscretzed doman Basc Concepts: 25

26 Some Examples of Real-World Fnte Element Dscretzatons Introducton: 26

27 FEM Termnology Element A geometrc sub-doman of the regon beng smulated, wth the property that t allows a unque (1) representaton of ts geometry and (2) dervaton of the approxmaton (nterpolaton) functons. Node A geometrc locaton n the element whch plays a role n the dervaton of the nterpolaton functons and t s the pont at whch soluton s sought. Mesh A collecton of elements (or nodes) that replaces the actual doman. Weak Form An ntegral statement equvalent to the governng equatons and natural boundary condtons. More to come. Basc Concepts: 27

28 FEM Termnology (contnued) Fnte Element Model A set of algebrac equatons relatng the nodal values of the prmary varables (e.g., dsplacements) to the nodal values of the secondary varables (e.g., forces) n an element. Fnte element model s NOT the same as the fnte element method. There s only one fnte element method but there can be more than one fnte element model of a problem (dependng on the approxmate method used to derve the algebrac equatons). Numercal Smulaton Evaluaton of the mathematcal model (.e., soluton of the governng equatons) usng a numercal method and computer. Basc Concepts: 28

29 Major Steps of Fnte Element Model Development Begn wth the governng equatons of the problem Develop ts weak form over a typcal fnte element Approxmate the soluton over each fnte element Obtan algebrac relatons among the quanttes of nterest over each fnte element (.e., fnte element model) Basc Concepts: 29

30 Major Steps of Fnte Element Model Development Engneerng Problem Formulaton Governng (Dfferental) Equatons Sold mechancs Weak Form Development Vrtual work statements Fnte Element Model Development Basc Concepts: 30